Download Phys 197 Homework Solution 41A Q3.

Phys 197 Homework Solution 41A
Q3.
For a body orbiting the sun, such as a planet, comet, or asteroid, is there any restriction
on the z-component of its orbital angular momentum such as there is with the z-component of the
electron’s orbital angular momentum in hydrogen? Explain.
————–
Yes. It must be a multiple of h̄. To give this some perspective, consider a 1-kg rock sharing
Earth’s orbit (R = 1.5×109 m, v = 3×104 m/s). L = mvr = (1 kg)(3×104 m/s)(1.5×109 m) =
4.5 × 1013 kg m2 /s. To get an ℓ value, divide this by h̄:
ℓ = L/h̄ = (4.5 × 1013 /(1.05 × 10−34 ) = 4.3 × 1047 .
Since the radius of the orbit scales as ℓ2 , the ratio of orbital radii is (ℓ + 1)2 /ℓ2 = 1 + 2/ℓ + 1/ℓ2 .
Neglect the 1/ℓ2 to get the fractional change as 2/ℓ = 4.6 × 10−48 . This means that the next
allowed orbit is farther out by a distance of (4.6 × 10−48 )(1.5 × 109 m) = 7 × 10−39 m.
Q4.
Why is the analysis of the helium atom much more complex than that of the hydrogen
atom, either in a Bohr type of model or using the Schrödinger equation?
————–
In calculating allowed orbits of the “second” electron, the electrostatic influence of the first
electron must be included. If the first electron is put into the lowest Schrödinger solution, that
is only a start, because the second electron affects the first, so that the first electron orbit
(state) is perturbed and must be re-claculated. Now, the wavefunction for the second must be
recalculated with the perturbation. Which further perturbs the first ....
Q7.
In the Stern-Gerlach experiment, why is it essential for the magnetic field to be inhomogeneous (that is, nonuniform)?
————–
A homogeneous field only precesses the angular momentum and magnetic moment vectors. The
field gradient (∂Bz /∂z) exerts a force on the dipole.
Q11.
Table 41.3 shows that for the ground state of the potassium atom, the outermost electron
is in a 4s state. What does this tell you about the relative energies of the 3d and 4s levels for this
atom? Explain.
————–
Electrons fill in order of energy. Therefore, the 4s shell is lower than the 3d shell when the 1s,
2s, 2p, 3s, and 3p are filled. If all electrons were are stripped away and one given back, it would
find the 3d to be lower in energy that the 4s. The presence of the lower electrons modifies this
relationship.
Q16.
A small amount of magnetic-field splitting of spectral lines occurs even when the atoms
are not in a magnetic field. What causes this?
————–
The spin-orbit coupling. One way to look at it is that the orbit creates a magnetic field, and
the electron spin interacts with this field.
Q23.
Can a hydrogen atom emit x rays? If so, how? If not, why not?
————–
No Way. The highest energy photon it can emit is
Eph = 13.6 eV ⇒ λ = (1240 eV · nm)/(13.6 eV) = 91 nm. This is still ultraviolet by
anyone’s band division.
P8.
An electron is in the hydrogen atom with n = 5 (a) Find the possible values of L and Lz
~ and the
for this electron, in units of h̄. (b) For each value of L, find all the possible angles between L
z-axis. (c) What are the maximum and minimum values of the magnitude of the angle between and
the z-axis?
————–
For n = 5, ℓ may take on the values 0,p
1, 2, 3, and 4.
L (in units of h̄) is calculated by L = ℓ(ℓ + 1)
Lz in units of h̄ is the same as ml which may take on the values −ℓ, −ℓ + 1, ... ℓ − 1, ℓ.
The angle is calculated as Cos−1 (Lz /L).
(a), (b), and (c) combined
ℓ
L
Lz
angle
max/min
0
0.0000
0
indeterminate
N/A
1
1.4142
-1
135.0
max for ℓ = 1
1
1.4142
0
90.0
1
1.4142
1
45.0
min for ℓ = 1
2
2.4495
-2
144.7
max for ℓ = 2
2
2.4495
-1
114.1
2
2.4495
0
90.0
2
2.4495
1
65.9
2
2.4495
2
35.3
min for ℓ = 2
3
3.4641
-3
150.0
max for ℓ = 3
3
3.4641
-2
125.3
3
3.4641
-1
106.8
3
3.4641
0
90.0
3
3.4641
1
73.2
3
3.4641
2
54.7
3
3.4641
3
30.0
min for ℓ = 3
4
4.4721
-4
153.4
max for ℓ = 4
4
4.4721
-3
132.1
4
4.4721
-2
116.6
4
4.4721
-1
102.9
4
4.4721
0
90.0
4
4.4721
1
77.1
4
4.4721
2
63.4
4
4.4721
3
47.9
4
4.4721
4
26.6
min for ℓ = 4
The orbital angular momentum of an electron has a magnitude of 4.716 × 10−34 kg m2 /s.
What is the angular-momentum quantum number for this electron?
P9.
————–
−34
Calculate how many times h̄ this L is; suppress the
√ common factor of 10 .
p4.716/1.054√= 4.474. This is close enough to 20 = 4.472 to say that ℓ = 4 (since L =
4(4 + 1) = 20).
P19.
A hydrogen atom in the 5g state is placed in a magnetic field of 0.600 T that is in
the z-direction. (a) Into how many levels is this state split by the interaction of the atom’s orbital
magnetic dipole moment with the magnetic field? (b) What is the energy separation between adjacent
levels? (c) What is the energy separation between the level of lowest energy and the level of highest
energy?
————–
(a) Recall that the g sublevel corresponds to ℓ = 4. The magnetic quantum number mℓ takes
on integer values from -4 to +4, so the splitting is into 9 levels.
(b) To be definite, take the splitting between mℓ = 1 and mℓ = 0. Using Eq 41.36:
∆U = (1 − 0)µB B = (9.27 × 10−24 J/T)(0.6 T) = 5.56 × 10−24 J = 3.47 × 10−5 eV.
(c) Since there are 9 levels, the difference from highest to lowest is 8 times the value above, or
2.78 × 10−4 eV.
You should take a look at the units, including the fact that the tesla can be written as 1 T =
kg/(C · s).
P21.
Classical Electron Spin. (a) If you treat an electron as a classical spherical object with
a radius of p
1.0 × 10−17 m, what angular speed is necessary to produce a spin angular momentum of
magnitude 3/4 h̄? (b) Use v = rω and the result of part (a) to calculate the speed v of a point at
the electron’s equator. What does your result suggest about the validity of this model?
————–
2
Take the electron’s
p moment of inertia as I = (2/5)mr . Then
Lspin = Iω = 3/4 h̄ ⇒
p
ω = 3/4 h̄/[(2/5)mr2 ] and
p
v = rω = 3/4 h̄/[(2/5)mr.
(a) ω = (0.866)(1.05 × 10−34 /[0.4 · (9.11 × 10−31 kg)(10−17 m)2 ] = 2.5 × 1030 rad/s.
(b) v = 2.5 × 1013 m/s = 105 c. Warp-warp-warp. Einstein disapproves.
My suggestion for the classical electron radius reduces the speed to a mere 9× 1010 m/s = 300 c.
Einstein’s frown didn’t change much.
The spin of the electron is simply some intrinsic property it has, which isn’t explainable with
any reasonable classical parameters.